Proposition 9

Let $A$ be a noetherian ring, $M$ and $N$ two A-modules of finite type, and $F$ the support of $N$. Then $\mathrm{prof}_F(M)$ is the greatest lower bound (in $N \cup \{+\infty\}$) of the set of integers $n$ such that $\mathrm{Ext}_A^n(N, M) \neq 0$.

By Remark 1, one has $\mathrm{Ext}_A^i(N, M) = 0$ for every $i < \mathrm{prof}_F(M)$. It remains to prove that if $\mathrm{prof}_F(M) = n < +\infty$, one has $\mathrm{Ext}_A^n(N, M) \neq 0$. Let $J$ be the annihilator of $N$; one has $F = V(J)$, hence $\mathrm{prof}_F(M) = \mathrm{prof}_A(J; M)$. By Cor. 1 of Th. 2 (No. 4), there exists an $M$-regular sequence $(x_1, \ldots, x_n)$ of length $n$ consisting of elements of $J$, and the depth relative to $J$ of the A-module $\overline{M} = M/(x_1 M + \ldots + x_n M)$ is zero. By A, X, p. 166, Prop. 9, it is enough to prove that $\mathrm{Hom}_A(N, \overline{M})$ is nonzero. Now, by Prop. 8, there exists $p \in \mathrm{Supp}(M) \cap \mathrm{Supp}(N)$ such that $\mathrm{prof}_{A_p}(\overline{M}_p) = 0$, that is to say $\mathrm{Hom}_{A_p}(\kappa(p), \overline{M}_p) \neq 0$. Since $N_p$ is nonzero, the $\kappa(p)$-vector space $N_p \otimes_{A_p} \kappa(p)$ is nonzero (Nakayama's lemma), and so is its dual; there therefore exists a surjective $A_p$-linear mapping of $N_p$ onto $\kappa(p)$. It follows that one has $\mathrm{Hom}_{A_p}(N_p, \overline{M}_p) \neq 0$, hence $\mathrm{Hom}_A(N, \overline{M}) \neq 0$ (II, § 2, No. 7, Prop. 19), which was to be proved.